Problem: A circle has a sector with area $5\pi$ and central angle $200^\circ$. What is the area of the circle? ${9\pi}$ $\color{#9D38BD}{200^\circ}$ ${5\pi}$
Answer: The ratio between the sector's central angle $\theta$ and $360^\circ$ is equal to the ratio between the sector's area, $A_s$ , and the whole circle's area, $A_c$ $\dfrac{\theta}{360^\circ} = \dfrac{A_s}{A_c}$ $\dfrac{200^\circ}{360^\circ} = 5\pi \div A_c$ $\dfrac{5}{9} = 5\pi \div A_c$ $A_c \times \dfrac{5}{9} = 5\pi$ $A_c = 5\pi \times \dfrac{9}{5}$ $A_c = 9\pi$